Similarity function

Given a set of ontological entities $ O $, a similarity function is a function $ \sigma:O\times O \longrightarrow \mathbb{R}^+ $ that associates to every pair of entities of $ O $, a real number that express the resemblance between the entities, and such that holds the following properties:

Non-negativity: $ \forall x,y \in O, \sigma(x,y)\geq 0 $

Maximality: $ \forall x,y \in O, \sigma(x,x)\geq \sigma(x,y) $

Some authors add to this list this other property:

Simmetry: $ \forall x,y\in O, \sigma(x,y) = \sigma(y,x) $

In the same way, it is possible to define the dual concept of dissimiarity function as a function $ \delta:O\times O \longrightarrow \mathbb{R}^+ $, that express the difference between two entities, with the following properties:

Non-negativity $ \forall x,y \in O, \sigma(x,y)\geq 0 $

Minimality: $ \forall x\in O, \delta(x,x)= 0 $

A (dis)similarity function is normalized if their range is the real interval $ [0,1] $.